Saturday, 14 January 2012

Lemniscate functions

The first time I encountered these weird objects of analysis was probably while surfing the book of E. Kamke which in turn gave a reference to Whittakker and Watson. I was at that time delving into the methods of analytic geometry and only new that lemniscate was an 8-shaped algebraic curve of the 4th order, a particular case of Cassini ovals. So I was pretty shocked to find out that it could give rise to some "trigonometric" system.

Being fresh from the first course on calculus, I attempted an investigation of the properties of lemniscate functions by means of only the very basic techniques used derive similar results for circular functions. I wanted to derive formulas for derivatives and primitives, addition theorems, complementary formulas, etc.

However, looking back at that paper I see that in the most crucial steps I just took the known relations for Jacobi elliptic functions (from W&W book) and then reduced them to the particular case of lemniscate functions.

I think that lemniscate functions can be used for changing variable when evaluating certain integrals, or converting differential equations into manageable forms. Although in my rather limited practice I have never encountered the cases where it would be appropriate, I still want to make a small, but independent account of these devices and keep them in my arsenal waiting for the right moment to come.

So, enough of the words, let's get down to business.

Lemniscate functions arise when rectifying the length of lemniscate and are defined by inversion of an integral (see Wolfram article for intermediate steps)

2 comments:

Could I suggest that instead of writing "sl\phi" in TeX code, you write "\operatorname{sl}\phi"? That will be prevent italicization (contrast the non-italicized "arcsin" above with the immediately following italicized "sl" and result in proper spacing before and after "sl" in expressions like "3\operatorname{sl}\phi". Spacing is conspicuously missing in some of the expressions above.